The present invention relates in general to electronic filter technology and in particular to complex filters implemented in integrated circuits.
Complex filters are useful in certain applications in wireless communications. Complex filters offer selective suppression of positive or negative frequency components of a complex or real signal. This feature of complex filters contrasts with the operation of real filters in that real filters have a transfer function that is symmetric around the direct current (DC) position. The ability of complex filters to suppress positive or negative frequency components enables the suppression of image frequencies of a signal. The suppression of image frequencies is a very important consideration in the design and operation of wireless transceivers.
A review of complex signals and complex filters will be useful to understand the present invention. A real signal may have both a positive frequency component and a negative frequency component. For example, a cosine signal cos (xcfx89t) equals (ejxcfx89+exe2x88x92jxcfx89)/2 and a sine signal sin (xcfx89t) equals (ejxcfx89+exe2x88x92jxcfx89)/2j. The letter j represents the square root of minus one. That is, j2=xe2x88x921. The letter j therefore represents one imaginary unit.
A complex signal is a signal that is composed of two real signals in which one of the real signals is multiplied by j. A complex signal therefore has the form:
x(t)=xr(t)+j xi(t)xe2x80x83xe2x80x83(1)
where xr(t) is a real signal that represents the real component of the complex signal x(t) and where xi(t) is a real signal that represents the imaginary component of the complex signal x(t).
A complex signal x(t) may be amplified by multiplication by a complex constant A+j B. For example, let y(t) be the result of multiplying the complex signal x(t) by the complex constant A+j B. Then
y(t)=(A+j B) x(t)xe2x80x83xe2x80x83(2)
where
y(t)=yr(t)+j yi(t).xe2x80x83xe2x80x83(3)
The expression Yr(t) represents a real signal that is the real component of the complex signal y(t) and the expression yi(t) represents a real signal that is the imaginary component of the complex signal y(t).
Substituting Equation (1) into Equation (2) and multiplying and equating the real and imaginary parts of the result with y(t) gives:
Yr(t)=A xr(t)xe2x88x92B xi(t)xe2x80x83xe2x80x83(4)
and
yi(t)=B xr(t)+A xi(t).xe2x80x83xe2x80x83(5)
Similarly, a complex signal x(t) may be multiplied by another complex signal z(t) where
z(t)=Zr(t)+j zi(t).xe2x80x83xe2x80x83(6)
The multiplication of x(t) by z(t) is represented by:
y(t)=z(t)xc2x7x(t)xe2x80x83xe2x80x83(7)
The result of multiplying x(t) by z(t) may be obtained by substituting Equation (6) into Equation (7) and multiplying and equating the real and imaginary parts of the result with y(t). The result is:
yr(t)=zr(t)xr(t)xe2x88x92zi(t)xi(t)xe2x80x83xe2x80x83(8)
and
yi(t)=Zr(t)xr(t)+zi(t)xi(t).xe2x80x83xe2x80x83(9)
Complex signals may be filtered by real filters or by complex filters. A real filter has a real impulse response hr(t). The transfer function Hr(jxcfx89) is a rational polynomial function of jxcfx89. The transfer function Hr(jxcfx89) can be real only if Hr(jxcfx89)=Hr*(xe2x88x92jxcfx89).
A complex filter has a complex impulse response
h(t)=hr(t)+j hi(t)xe2x80x83xe2x80x83(10)
and a complex transfer function
H(jxcfx89)=Hr(jxcfx89)+j Hi(107).xe2x80x83xe2x80x83(11)
The response of a linear time invariant system to an arbitrary input x(t) can be expressed as the convolution of x(t) and the impulse response h(t) of the system. That is,
y(t)=h(t)◯x(t)xe2x80x83xe2x80x83(12)
where the symbol ◯ represents the convolution operation. Applying the well known time convolution theorem of the Fourier transform to Equation (12) gives:
Y(jxcfx89)=H(jxcfx89)xc2x7x(jxcfx89)xe2x80x83xe2x80x83(13)
where Y(jxcfx89) is a complex output signal and X(jxcfx89) is a complex input signal. H(jxcfx89) is a rational complex polynomial that is a function of jxcfx89.
Because the input signal X(jxcfx89) is complex, X(jxcfx89) is composed of a real part and an imaginary part.
X(jxcfx89)=Xr(jxcfx89)+j Xi(jxcfx89).xe2x80x83xe2x80x83(14)
where Xr(jxcfx89) represents the real part of X(jxcfx89) and where Xi(jxcfx89) represents the imaginary part of X(jxcfx89). Similarly, because the output signal Y(jxcfx89) is also complex, Y(jxcfx89) is also composed of a real part and an imaginary part.
Y(jxcfx89)=Yr(jxcfx89)+j Yi(jxcfx89).xe2x80x83xe2x80x83(15)
where Yr(jxcfx89) represents the real part of Y(jxcfx89) and where Yi(jxcfx89) represents the imaginary part of Y(jxcfx89).
Substituting Equations (11), (14) and (15) into Equation (13) and multiplying and equating the real and imaginary parts of the result to the real and imaginary parts of Y(jxcfx89) gives:
Yr(jxcfx89)=Hr(jxcfx89)xe2x88x92Xr(jxcfx89)xe2x88x92Hi(jxcfx89)Xi(jxcfx89)xe2x80x83xe2x80x83(14a)
Yi(jxcfx89)=Hr(jxcfx89)Xr(jxcfx89)+Hi(jxcfx89)Xi(jxcfx89)xe2x80x83xe2x80x83(15a)
In the time domain Equations (14a) and (15a) give:
yr(t)=hr(t)◯xr(t)xe2x88x92hi(t)◯xi(t)xe2x80x83xe2x80x83(16)
yi(t)=hr(t)◯xr(t)+hi(t)◯xi(t)xe2x80x83xe2x80x83(17)
where the symbol ◯ represents the convolution operation.
The equation of a transfer function having a complex pole has the form:                               H          ⁡                      (                          j              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                  A                      s            +                          (                              p                ±                                  j                  ⁢                                      xe2x80x83                                    ⁢                  q                                            )                                                          (        18        )            
The letter A represents a constant. The letter s represents the quantity jxcfx89. The letter p represents the real part of the complex pole and the letter q represents the imaginary part of the complex pole. Substitution of Equation (18) into Equation (13) gives:                               Y          ⁡                      (                          j              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                              A                          s              +                              (                                  p                  ±                                      j                    ⁢                                          xe2x80x83                                        ⁢                    q                                                  )                                              ·                      X            ⁡                          (                              j                ⁢                                  xe2x80x83                                ⁢                ω                            )                                                          (        19        )            
One of the main applications for complex filters is the selective suppression of positive or negative frequency components of a complex or real signal. This may be accomplished by using a bandpass filter that is obtained from the linear frequency transformation of a lowpass filter. A complex lowpass filter that is centered on the direct current (DC) value (i.e., jxcfx89=0) of the jxcfx89 axis of a H(jxcfx89)/jxcfx89 plane may be linearly transformed to create a complex bandpass filter that is centered on another value, jxcfx89c, of the jxcfx89 axis.
Using the linear transformation
s=jxcfx89xe2x88x92jxcfx89cxe2x80x83xe2x80x83(20)
will result in a bandpass filter that has the form of the lowpass filter but is centered around the frequency xcfx89c. This form of bandpass filter has only the frequency shifted lowpass filter characteristics for positive frequencies. The transfer function of this form of bandpass filter suppresses negative frequency components.
Substituting Equation (20) into Equation (19) leads to the following design equations for the real and imaginary parts of the output signal Y(jxcfx89). The argument jxcfx89 in the expressions Y(jxcfx89) and X(jxcfx89) in Equations (21) and (22) will be omitted for clarity.                               Y          r                =                                            X              r                        -                                                            ω                  C                                A                            ⁢                              Y                i                                      -                                                            p                  A                                ⁢                                  Y                  r                                            ±                                                q                  A                                ⁢                                  Y                  i                                                                          j            ⁢                          xe2x80x83                        ⁢                          ω              A                                                          (        21        )                                          Y          i                =                                            X              i                        -                                                            ω                  C                                A                            ⁢                              Y                r                                      -                                                            p                  A                                ⁢                                  Y                  i                                            ∓                                                q                  A                                ⁢                                  Y                  r                                                                          j            ⁢                          xe2x80x83                        ⁢                          ω              A                                                          (        22        )            
It would be desirable to provide circuitry on an integrated circuit that is capable of implementing a complex filter of the type represented by a transfer function having a complex pole. In particular, it would be desirable to provide an apparatus for providing the real part Yr(jxcfx89) and imaginary part Yi(jxcfx89) of an output signal Y(jxcfx89) that results from multiplying an input signal X(jxcfx89) by a transfer function H(jxcfx89) that has a complex pole.
The apparatus of the invention comprises circuitry that is capable of implementing a complex filter of the type that is represented by a transfer function having a complex pole. The apparatus creates real and imaginary parts, Yr(jxcfx89) and Yi(jxcfx89), of a complex output signal from real and imaginary parts, Xr(jxcfx89) and Xi(jxcfx89), of a complex input signal by implementing a transfer function that has a complex pole. The apparatus generally comprises an operational amplifier circuit and an input circuit comprising a plurality of input resistors. The input resistors in the input circuit may be variable and may be tuned to adjust various operating parameters of the complex filter.
It is an object of the present invention to provide an apparatus for implementing a complex filter of the type that is represented by a transfer function having a complex pole.
It is also an object of the present invention to provide an apparatus for providing the real part Yr(jxcfx89) and imaginary part Yi(jxcfx89) of an output signal Y(jxcfx89) that results from multiplying an input signal X(jxcfx89) by a transfer function H(jxcfx89) that contains a complex pole.
It is another object of the present invention to provide an apparatus for implementing a complex filter where the apparatus comprises a plurality of variable resistors that may be tuned to adjust operating parameters of the complex filter.
It is a further object of the present invention to provide an apparatus for implementing a complex filter where the apparatus comprises a plurality of variable resistors in which each resistor may be independently tuned to adjust operating parameters of the complex filter.
It is yet another object of the present invention to provide an apparatus for implementing a complex filter where the apparatus comprises a plurality of variable resistor pairs in which each resistor pair may be independently tuned to adjust operating parameters of the complex filter.
Other objects and advantages of the invention will become apparent as the description proceeds.
The foregoing has outlined rather broadly the features and technical advantages of the present invention so that those skilled in the art may better understand the Detailed Description of the Invention that follows. Additional features and advantages of the invention will be described hereinafter that form the subject matter of the claims of the invention. Those skilled in the art should appreciate that they may readily use the conception and the specific embodiment disclosed as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. Those skilled in the art should also realize that such equivalent constructions do not depart from the spirit and scope of the invention in its broadest form.
Before undertaking the Detailed Description of the Invention, it may be advantageous to set forth definitions of certain words and phrases used throughout this patent document: The terms xe2x80x9cincludexe2x80x9d and xe2x80x9ccomprisexe2x80x9d and derivatives thereof, mean inclusion without limitation, the term xe2x80x9corxe2x80x9d is inclusive, meaning xe2x80x9cand/orxe2x80x9d; the phrases xe2x80x9cassociated withxe2x80x9d and xe2x80x9cassociated therewith,xe2x80x9d as well as derivatives thereof, may mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, to bound to or with, have, have a property of, or the like; and the term xe2x80x9ccontroller,xe2x80x9d xe2x80x9cprocessor,xe2x80x9d or xe2x80x9capparatusxe2x80x9d means any device, system or part thereof that controls at least one operation. Such a device may be implemented in hardware, firmware or software, or some combination of at least two of the same. It should be noted that the functionality associated with any particular controller may be centralized or distributed, whether locally or remotely. Definitions for certain words and phrases are provided throughout this patent document. Those of ordinary skill should understand that in many instances (if not in most instances), such definitions apply to prior, as well as future uses of such defined words and phrases.